The solution of the asymmetric persistent random walk of photons
in one dimension
appears to be identical to the
solution of the symmetric process observed in a translation frame.
The persistent random walk as a point process in space is not Markovian,
but this flaw can be cured by considering it as a point process in
phase space. Chapman-Kolmogorov's relations are
satisfied only when considering the solution with pure initial
conditions in phase space.
Remarkably, the solution of the persistent random walk
in two dimensions, considered as a process in phase space, displays the
same properties as the one-dimensional process, confirming the relevance
of the concept of phase space process.
One should naturally expect
the same properties for the solution of the three-dimensional persistent random
walk in phase space, which would be of interest in astrophysics
for instance, but even an exact solution as a process in space
is not known (and often claimed not to exist [1]}).
It is however possible that a solution in phase space exists.
Possibly, such a solution is expressed in terms of a
Lorentz invariant variable such as \(\mathcal {X}\)  (which three-dimensional
counterpart also is Lorentz invariant).
In any event, I believe that this
Letter will trigger progress in this direction.
